The maximal D-compact extension of a completely regular space is constructed, and this procedure is used to construct the unique D-compact group extension of a totally bounded topological group. Sufficient conditions are found for every power of a topological group to be countably compact. In this paper, those Hausdorff spaces, all of whose powers are countably compact, are characterized, and partial results on the corresponding question for pseudocompactness are obtained. Ultrafilters on ?.- The Rudin-Keisler Order on ?(?) and the Canonical Function ?:?(? x ?) ?(?) x ?(?).- The Rudin-Frolik Order.- Non-Homogeneity of Certain Spaces.- Notes for 16.- Index of Symbols. Spaces Homeomorphic to (2?)?.- The Topological Characterization of (2?)?.- The Baire Category Properties of (2?)?.- Spaces of Ultrafilters Homeomorphic to Spaces (2?)?.- Applications to the Growth Spaces ?X\X.- Notes for 15.- 16. Topology of Spaces of Ultrafilters.- Certain Properties of F?-Spaces.- The Space of ?-Uniform Ultrafilters on ?.- Spaces of Uniform Ultrafilters and Homogeneous-Universal Boolean Algebras.- The Space of Sub-Uniform Ultrafilters.- Relations to Measurable Cardinals.- Notes for 14.- 15. Saturation of Ultraproducts.- Ultraproducts Modulo Regular Ultrafilters.- Ultraproducts Modulo Good Ultrafilters.- Shelah's Characterisation of Elementary Equivalence.- Characterisation of the Rudin-Keisler Order.- Notes for 13.- 14. Families of Almost Disjoint Sets.- Cardinalities of Families of Almost Disjoint Sets.- The Balcar-Vop?nka Theorem.- Cardinalities of Ultraproducts.- Notes for 12.- 13. Good Ultrafilters.- Families of Large Oscillation Modulo Filters the Fundamental Existence Theorem of Good Ultrafilters.- Additional Existence Results.- Directedness Properties of the Rudin-Keisler Order.- Adequate Ultrafilters on Special Boolean Algebras.- Notes for 10.- 11. The Rudin-Keisler Order on Types of Ultrafilters.- The Rudin-Keisler Order.- Rudin-Keisler Minimal Types in ?.- Good Ultrafilters.- Notes for 9.- 10. Large Cardinals.- Weakly Compact Cardinals: Combinatorial Equivalences.- Weakly Compact Cardinals: Boolean-Algebraic and Topological Equivalences.- Measurable Cardinals.- Descendingly Incomplete Ultrafilters.- Notes for 8.- 9. Basic Facts on Ultrafilters.- Notes for 7.- 8. The Jonsson Class of Boolean Algebras.- The Stone Space of the Homogeneous-Universal Boolean Algebras.- Properties of the Space S?.- Notes for 6.- 7. The Jonsson Class of Ordered Sets.- Notes for 5.- 6. The General Theory of Jonsson Classes.- Notes for 4.- 5. Intersection Systems and Families of Large Oscillation.- Intersection Systems and the Souslin Number.- Families of Large Oscillation.- Notes for 3.- 4.
Topology and Boolean Algebras.- Topology.- Finitary Properties of Boolean Algebras.- Stone's Duality.- The Completion of a Boolean Algebra and the Gleason Space of a Compact Space.- Notes for 2.- 3. Set Theory.- Ordinals.- Cardinal Arithmetic.- Notes for 1.- 2.
The properties of filter bases can be defined as follows: (1) they can be sequentially compact, wherein every countably filter base has a finer countable filter base that is convergent (2) they can be countably compact, wherein every countable filter base has an adherent point, (3) they can be ω-bounded, in which every filter base on a countable set has an adherent point, and (4) they can be totally countably compact, wherein every countable filter base has a finer countable filter base which is total.ġ. A space X is called ω-bounded if for every sequence f in X, the range of f is contained in a compact subset of X. A space X is called totally countably compact if every sequence f in X has a subsequence f | A whose range is contained in a compact subset of X. A space X is called countably compact if sequence in X has a cluster point. A space X is called sequentially compact if every sequence in X has a convergent subsequence. The chapter presents the basic theory, examples, and techniques of countable compactness and sequential compactness. We also prove that any product ∏ακH0(Xα) embeds into H(F(Xα:ακ)). We use the Frolík sum to produce countably compact spaces with additional properties (like first countability) whose hyperspaces are not countably compact.
We show, by assuming the existence of $\mathfrak c$-many selective ultrafilters, that there exists a subspace of $\beta \omega$ that is $(\kappa, \omega^*)$-pseudocompact for all $\kappa<\mathfrak c$, but $\text. We study the relations between a generalization of pseudocompactness, named $(\kappa, M)$-pseudocompactness, the countably compactness of subspaces of $\beta \omega$ and the pseudocompactness of their hyperspaces.